Oblique decision trees in transformed spaces.

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Thesis Discipline

Statistics

Degree Grantor

University of Canterbury

Degree Level

Doctoral

Degree Name

Doctor of Philosophy

Decision trees (DTs) play a vital role in statistical modelling. Simplicity and interpretability of the solution structure have made the method popular in a wide range of disciplines. In data classification problems, DTs recursively partition the feature space into disjoint sub-regions until each sub-region becomes homogeneous with respect to a particular class. Axis parallel splits, the simplest form of splits, partition the feature space parallel to feature axes. However, for some problem domains DTs with axis parallel splits can produce complicated boundary structures. As an alternative, oblique splits are used to partition the feature space potentially simplifying the boundary structure. Various approaches have been explored to find optimal oblique splits. One approach is based on optimisation techniques. This is considered the benchmark approach, however, its major limitation is that the tree induction algorithm is computationally expensive. On the other hand, split finding approaches based on heuristic arguments have gained popularity and have made improvements on benchmark methods. This thesis proposes a methodology to induce oblique decision trees in transformed spaces based on a heuristic argument.

As the first goal of the thesis, a new oblique decision tree algorithm, called HHCART (\underline{H}ouse\underline{H}older \underline{C}lassification and \underline{R}egression \underline{T}ree) is proposed. The proposed algorithm utilises a series of Householder matrices to reflect the training data at each non-terminal node during the tree construction. Householder matrices are constructed using the eigenvectors from each classes' covariance matrix. Axis parallel splits in the reflected (or transformed) spaces provide an efficient way of finding oblique splits in the original space. Experimental results show that the accuracy and size of the HHCART trees are comparable with some benchmark methods in the literature. The appealing features of HHCART is that it can handle both qualitative and quantitative features in the same oblique split, conceptually simple and computationally efficient.

Data mining applications often come with massive example sets and inducing oblique DTs for such example sets often consumes considerable time. HHCART is a serial computing memory resident algorithm which may be ineffective when handling massive example sets. As the second goal of the thesis parallel computing and disk resident versions of the HHCART algorithm are presented so that HHCART can be used irrespective of the size of the problem.

HHCART is a flexible algorithm and the eigenvectors defining Householder matrices can be replaced by other vectors deemed effective in oblique split finding. The third endeavour of this thesis explores this aspect of HHCART. HHCART can be used with other vectors in order to improve classification results. For example, a normal vector of the angular bisector, introduced in the Geometric Decision Tree (GDT) algorithm, is used to construct the Householder reflection matrix. The proposed method produces better results than GDT for some problem domains. In the second case, \textit{Class Representative Vectors} are introduced and used to construct Householder reflection matrices. The results of this experiment show that these oblique trees produce classification results competitive with those achieved with some benchmark decision trees.

DTs are constructed using two approaches, namely: top-down and bottom-up. HHCART is a top-down tree, which is the most common approach. As the fourth idea of the thesis, the concept of HHCART is used to induce a new DT, HHBUT, using the bottom-up approach. The bottom-up approach performs cluster analysis prior to the tree building to identify the terminal nodes. The use of the Bayesian Information Criterion (BIC) to determine the number of clusters leads to accurate and compact trees when compared with Cross Validation (CV) based bottom-up trees. We suggest that HHBUT is a good alternative to the existing bottom-up tree especially when the number of examples is much higher than the number of features.